The Stability Threshold for 3D MHD Equations Around Couette with Rationally Aligned Magnetic Field

In this article, we consider regularity of solutions to the 3D viscous MHD equations. Regularity criteria are established in terms of the pressure or the gradient of pressure, which improve the results in Y. Zhou [Regularity criteria for the 3D MHD equations in terms of the pressure, Int. J. Non-Linear Mech. 41(10) (2006), pp. 1174–1180] where additional conditions on the magnetic field are also needed.

This paper considers the global regularity to the 3D incompressible MHD equations with large initial data in bounded domains. Let μ, ν, u, and b denote the viscosity coefficient, magnetic diffusivity, velocity field, and magnetic field, respectively. We construct new systems for (u − b) and (u + b) to overcome the difficulties caused by the large initial data. It is shown that (u,b)H1 is globally bounded as long as (u0−b0)H1+μ−ν(μ+ν)−1 or (u0+b0)H1+μ−ν(μ+ν)−1 is sufficiently small, which indicates that the Navier-Stokes equations can be regularized by the magnetic field.

In this paper, we prove some logarithmically improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain.

In this note, we give a new proof to the energy conservation for the weak solutions of the incompressible 3D MHD equations. Moreover, we give the lower bounds for possible singular solutions to the incompressible 3D MHD equations.

In this paper, we study the regularity criterion of weak solutions to the three-dimensional (3D) MHD equations. It is proved that the solution $(u,b)$ becomes regular provided that one velocity and one current density component of the solution satisfy% \begin{equation} u_{3}\in L^{\frac{30\alpha }{7\alpha -45}}\left( 0,T;L^{\alpha ,\infty }\left( \mathbb{R}^{3}\right) \right) \text{ \ \ \ with \ \ }\frac{45}{7}% \leq \alpha \leq \infty , \label{eq01} \end{equation}% and \begin{equation} j_{3}\in L^{\frac{2\beta }{2\beta -3}}\left( 0,T;L^{\beta ,\infty }\left( \mathbb{R}^{3}\right) \right) \text{ \ \ \ with \ \ }\frac{3}{2}\leq \beta \leq \infty , \label{eq02} \end{equation}% which generalize some known results.

Regularity criteria for the 3D MHD equations via one directional derivative of the pressure

Global stability of large solutions to the 3D nonhomogeneous incompressible MHD equations

We consider the problem of regularization by noise for the three dimensional magnetohydrodynamical (3D MHD) equations. It is shown that, in a suitable scaling limit, multiplicative noise of transport type gives rise to bounds on the vorticity fields of the fluid velocity and magnetic fields. As a result, if the noise intensity is big enough, then the stochastic 3D MHD equations admit a pathwise unique global solution for large initial data, with high probability.

Preface. 1: Accuracy. Progress in Applied Numerical Analysis for Computational Fluid Dynamics M.B. Giles. Examples of Error Propagation from Discontinuities B. Engquist, B. Sjogreen. Accurate Finite Difference Algorithms J.V. Goodrich. Computational Considerations for the Simulation of Discontinuous Flows M. H. Carpenter, J.H. Casper. Space-Time Methods for Hyperbolic Conservation Laws R.B. Lowrie, et al. II: Boundary Conditions and Stiffness Issues. Anisotropic Mesh Adaptation: A Step Towards a Mesh-Independent and User-Independent CFD W.G. Habashi, et al. Artificial Boundary Conditions for Infinite-Domain Problems S.V.Tsynkov. Issues and Strategies for Hyperbolic Problems with Stiff Source Terms M. Arora, P.L. Roe. III: Discontinuities. Numerical Methods for a One-Dimensional Interface Separating Compressible and Incompressible Flows R. Fedkiw, et al. On Some Outstanding Issues in CFD (1996) R. Lohner. Accurate and Robust Methods for Variable Density Incompressible Flows with Discontinuities W.J. Rider, et al. A Variational Approach to Deriving Smeared-Interface Surface Tension Models D. Jacqmin. IV: Other Applications. Compounded of Many Simples: Reflections on the Role of Model Problems in CFD P. Roe. A Unified CFD-Based Approach to a Variety of Problems in Computational Physics R.K. Agarwal. Second Order Godunov Schemes for 2D and 3D MHD Equations and Divergence-Free Condition Wenlong Dai, P.R. Woodward. On Multidimensional Positively Conservative High-Resolution Schemes T. Linde, P.L. Roe. A New Scheme for the Solutions of Multidimensional MHD Equations N. Aslan, T. Kamash . V: Convergence. Local Preconditioning: Who Needs It? B. Van Leer. The Quest for Diagonalization of Differential Systems P. Roe, E. Turkel. Multidimensional Upwinding: Unfolding the Mystery D. Sidilkover.

Numerical simulation of the convective plasma dynamics stage at the ionosphere motion by means of 3D MHD equations

In this paper, we study the 3D compressible non-isentropic MHD equations with zero resistivity in a bounded domain. We first establish a regularity criterion for the strong solutions, and then, we combine it with an abstract bootstrap argument to prove the global well-posedness of the system under some assumptions on the viscosity coefficients and the initial data. Our results do not need the positivity of initial density, and thus, it may vanish in an open subset of the domain.

On the 3D ideal MHD equations with partial damping

In this paper, we consider the blow-up of smooth solutions to the 3D ideal MHD equations. Let (u, b) be a smooth solution in (0, T). It is proved that the solution (u, b) can be extended after t = T if $$ {\left( {\nabla \times u,\nabla \times b} \right)} \in L^{1} {\left( {0,T;\ifmmode\expandafter\dot\else\expandafter\.\fi{B}^{0}_{{\infty ,\infty }} } \right)} $$ . This is an improvement of the result given by Caflisch, Klapper, and Steele [3].

In this work, we present a uniqueness condition depending on (ν, μ, s, u0, B0, F) and consider the fully discrete Galerkin method for the 3D time-dependent incompressible MHD equations. Furthermore, we provide the global uniform well-posedness (or the existence, uniqueness and uniform stability of the solution) of the 3D time-dependent incompressible MHD equations under the uniqueness condition by use of the compact theorem and a new a priori estimate of the fully discrete Galerkin solution.

Stabilizing effect of the magnetic field and decay estimates for the 3D MHD equations with only one direction dissipation